Since there is no exact rarefaction wave profile for the
Navier-Stokes equations (REF ), the following approximate
rarefaction wave profile satisfying the Euler equations was
motivated by Matsumura-Nishihara [1]} and Xin [2]}.
For the completeness of the presentation, we include its definition
and the properties listed in Lemma REF . However, Lemma
REF  is a little different from [2]} as stated after
Lemma REF .
